Con - Massachusetts Institute of Technology · Con ten ts In tro duction Theory of electronic...
Transcript of Con - Massachusetts Institute of Technology · Con ten ts In tro duction Theory of electronic...
Contents
Introduction ��
� Theory of electronic structure ��
��� The Schr�odinger equation ��� � � � � � � � � � � � � � � � � � � �
����� Variational formulation � � ��� � � � � � � � � � � � � � � �
����� The Hellmann�Feynman theorem ��� � � � � � � � � � � �
��� Density Functional Theory � � � � � � � � � � � � � � � � � � � � ��
����� The Hohenberg�Kohn theorems � � � � � ��� � � � � � � �
����� The Levy approach � ��� � � � � � � � � � � � � � � � � � �
���� The representability problem � � � � � � � � � � � � � � � ��
����� Density op erators ��� � � � � � � � � � � � � � � � � � � � �
����� Canonical�ensemble formulation � � � � � � � � � � � � � �
�� The Kohn�Sham mapping �� � � � � � � � � � � � � � � � � � � �
���� Fractional occupancies Janak�s theorem ��� � � � � � � �
���� Finite�temperature extension �� � � � � � � � � � � � � �
� Methods of electronic structure ��
��� The Local Density Approx i m a t i o n � � � � � � � � � � � � � � � � �
��� The pseudopotential approximation � � � � � � � ��� � � � � � � �
�
� CONTENTS
����� Basis representation ��� � � � � � � � � � � � � � � � � � �
����� Smoothness and transferability ��� � � � � � � � � � � � �
����� The Kleinman�Bylander representation � � � � � � � � � ��
��� Minimizations and dynamics � � � � � � ��� � � � � � � � � � � � �
����� Car�Parrinello molecular dynamics �� � � � � � � � � � �
����� Conjugate�gradient minimization �� � � � � � � � � � � �
����� Metallic systems ��� � � � � � � � � � � � � � � � � � � � �
� Ensemble Density Functional Theory ��
��� Ensemble free energy � � � � � � � � � � � � � � � � � � � � � � � �
��� Minimization strategy ��� � � � � � � � � � � � � � � � � � � � � �
����� Doubly�preconditioned all�bands CG ��� � � � � � � � � �
����� Occupation matrix� an iterative s c heme � � � � � � � � � ��
��� Applications and comparisons � � � � � � � � � � � � � � � � � � �
� � � � ������ Self�consistency and molecular dynamics � � � �
� � � � � � � � � � � � � � � � � � � ������ Ionic relaxations � � �
�� Further development s � � � � � � ��� � � � � � � � � � � � � � � � �
� Generalized entropy and cold smearing ���
�� Generalized free energy � � � � � � � � � � � � � � � � � � � � � � ��
�� Smeared functionals � � � � � � � � � � � � � ���� � � � � � � � � � �
� � � � � � � � � � � � � � � � � ��� Entropy corrections � � � � � � � � �
���� Energy � � � � � � � � � � � � � � � � � � � � � � � � � � � ���
� � � � � � � � � � � ����� Forces and other derivatives � � � � � �
� Cold smearing � � � � � � � � � � � � � � � � � � � � � � � � � � � ��
CONTENTS �
� Ensemble�DFT molecular dynamics ���
��� Molecular dynamics simulations � � � � � � � � � � � � � � � � � ���
����� Technical details � � � �� � � � � � � � � � � � � � � � � � � �
��� Metal surfaces at �nite temperature � � � � � � � � � � � � � � � ���
��� Computational LEED � � � �� � � � � � � � � � � � � � � � � � � � �
����� Bulk Al � � � �� � � � � � � � � � � � � � � � � � � � � � � � �
����� Al���� thermal relaxations � � � � � � � � � � �� � � � � � �
����� Al���� layer�resolved displacement s � � � �� � � � � � � � �
����� Al���� � � � �� � � � � � � � � � � � � � � � � � � � � � � � �
��� Sampling the phase space � � � �� � � � � � � � � � � � � � � � � � �
����� Preroughening di�usion and liquid layering � � � � � � ���
Conclusions ���
Acknowledgments ���
Bibliography ���
Chapter �
Theory of electronic structure
Introduction
A brief review is given of the problem of �nding the ground state for a system
of interacting electrons immersed in an external potential� The fundamental
law o f q u a n tum mechanics�the Schr�odinger equation ����is introduced� and
its equivalence to a variational principle is established� A modern and ex�
tremely successful reformulation of the problem�that goes under the name
of Density F unctional Theory ���is presented� and its intrinsic nature as
a minimization problem is highlighted� The role of temperature �� and the
introduction of statistical operators to describe mixed states is discussed�
Finally� the problem is recast via the Kohn�Sham ��� mapping onto a system
of non�interacting electrons in a self�consistent external potential�
�
�� CHAPTER �� THEORY O F E L E C T R ONIC STRUCTURE
��� The Schr�odinger equation
At the present state of human knowledge� quantum mechanics can
be regarded as the fundamental theory of atomic phenomena�
�L� I� Schi�� Quantum Mechanics� �����
This statement has been supported since the early days that followed the
introduction of quantum mechanics by a n e v er growing evidence for the ac
curacy of its predictions when compared with experiments� Notwithstanding
the existence of many unresolved questions in the formal interpretation of the
theory ���� the range of its applications has extended from the description of
single isolated atoms to all areas of fundamental and applied physics and
chemistry� As a simple example of its extreme accuracy� the gyromagnetic
ratio for the electron can be determined in agreement with experiment with
a precision greater than one part in a billion� Other disciplines� ranging from
materials science to biology� are starting now to benet from this predic
tive p o wer and from the development o f v ery powerful electronic structure
methods� as described in Chapter �� that allow for the rst time to compute
fundamental properties of complex systems from rst principles� using quan
tum mechanics to determine the behaviour of their fundamental constituents�
electrons and ions�
In the following we are concerned with the groundstate electronic prop
erties of a nite� isolated system of N interacting electrons in an external
potential� The external potential considered is that generated by a congu
ration of atomic nuclei� assumed for the time being to be xed point c harges�
Some of these conditions will be relaxed or modied along the way� namely
���� THE SCHR
�ODINGER EQUATION ��
by i n troducing pseudopotentials ������ to describe the interactions between
the valence electrons and the rest of the system� by a l l o wing for the ions
to move adiabatically ��� in the �eld of the electrons ground state� and by
introducing periodic boundary conditions to reduce or eliminate �nitesize
errors�
The nonrelativistic timeindependent S c hr�odinger equation for the sys
tem described is
H��r1� � � � � rN
� � E��r1� � � � � rN
�� �����
where the Hamiltonian operator1 H is
H � Te
� Vn�e
� Ue�e
� Wn�n
�
� � X ��X
�
�
ri
2 � v�ri�
� � � Wn�n
������ � jri
� rj
ji i6�j
with X Z�
X Z�Z��
v�r� � �jr � R�j
and Wn�n
� � ������ R��
� ��
The terms in ����� are associated� respectively� with the kinetic energy Te
of the electrons� the potential energy Vn�e
of the electrons in the �eld of �
nuclei of charge Z�� the electrostatic energy Ue�e
be t ween the electrons� and
the electrostatic energy Wn�n
be t ween the nuclei�
Equation ����� is an eigenvalue equation for the N electron many bod y
wavefunction ��r1� � � � � rN
� � r1
� � � rN
being the spatial coordinates for the
electrons�� where H is Hermitian� in order for the solutions to be acceptable
they must belong to the Hilbert space L2 �R
3� � � � � � L
2 �R
3� of square
integrable functions� The spin coordinates will be ignored �but not the
spin���� nature of the electrons�� while some relativistic e�ects�relevant
1Atomic units are used� ~ � me
� e � � �� 0
� ��
�� CHAPTER �� THEORY O F E L E C T R ONIC STRUCTURE
for heavy atoms�can be taken into account a posteriori when constructing
the pseudopotential� The Fermi�Dirac statistics for the electrons enters the
problem as a restriction on the Hilbert space� requiring � to be antisymmet�
ric under the exchange of the coordinates of two electrons� This� translating
into the Pauli principle� accounts ultimately for the stability of matter� with
the ground state energy bounded below by a constant tim es the �rst power
of the particle number �����
The solutions to equation ���� form a complete set f�n
g ���� once nor�
malized� any t wo of them that correspond to di�erent eigenvalues are or�
thogonal� and the set of eigenvectors f�n
g is thus always considered fully or�
thonormal� According to the postulates of quantum mechanics ���� to each
�observable A there corresponds a Hermitian operator A and consequently its
complete set of orthonormal eigenfunctions f�ng with eigenvalues fang� the
wavefunction of the system can then always be expressed in terms of these
eigenfunctions as
X
� � cn
�n� ����
n
If we h a ve an ensemble of systems identically prepared� each measurement
will yield the eigenvalue an
with probability jcn
j2 � The expectation value of
A can thus be expressed� using Dirac notation� as
XX X�� � �
0hAi � h�jAj�i � cn0
cn
h�n
jAj�ni � jcn
j2 an� ����
n n0 n
R
��where h�jAj�i stands for ��A� dr1
� � � d rN
�
���� THE SCHR
�ODINGER EQUATION ��
����� Variational formulation
Let us consider a time�independent Hamiltonian� as that described in ������
for simplicity� w e assume that the system is either enclosed in a box of nite
size� or that periodic boundary conditions2 are applied� Under such con�
ditions� the �non�empty� spectrum of eigenvalues fEn
g and eigenfunctions
f�n
g is discrete3 �see Chap� of Ref� ��� for a discussion�� For an arbitrary
2function in the Hilbert space L2 �R
3� � � � � �L �R
3� that has non�vanishing
norm� we dene the functional E� � as
�h jH j i
E� � � �����h j i
The following theorem is easily proved�
�Variational Principle The Schr�odinger equation H � E for the Hamil�
�tonian H is equivalent to the variational principle �E � � � ��
Proof� Let us consider a generic unconstrained � taking the variation
��E� � h j i� in ����� we h a ve�
� ��E � � h j i � E� � h� j i � E� � h j� i � h� jHj i � h jH j� i�
2In electronic structure calculations it is useful to adopt a looser form of periodic
boundary conditions that goes under the names of Born and von Karman� ��r � Ni
ai
� �
��r�� i � � � �� � where ai
are the primitive v ectors of the cell and Ni
are the independent
primitive cells in each direction These conditions impose the existence of a discrete �nite
numbe r o f a llo wed values for the Bloch w ave v ector �a good quantum numb e r in a p e r io d ic
solid� and its corresponding energy The spacing between di�erent w avevectors goes to
zero in the limit Ni
�� r e c o vering thus the continuum limit for the spectrum of a truly
in�nite periodic crystal
3The discussion can be generalized to an Hamiltonian with an additional continuum
spectrum fen
g such that inffen
g � minfEng
�� CHAPTER �� THEORY O F E L E C T R ONIC STRUCTURE
�The following equivalences are established� exploiting the hermiticity o f H �
� � ��E ��� � � h ��j H � Ej �i h �j H � Ej ��i � � �H � E����j �i �
QED
The variational principle can be reformulated in order to include automat
ically the wavefunction normalization� by i n troducing an undetermined La
grange multiplier E� the constrained variational problem
���h �j H j �i � � � h �j �i � �
is thus transformed in the unconstrained variational problem
���h �j Hj �i � Eh �j �i � � � �����
The importance of the functional ����� is that it provides a variational bound
to the ground state energy of the system� this can be seen by expanding �
in the energy eigenvectors f �ng �
P P X j anj
2En
j an
j
2�En
� E0
�n� � an
�n
� E��� �
P
j an
j
2
� E��� � E0
�
n P
j an
j
2
nn n
where we h a ve indicated with E0
� minf En
g the ground state energy� I t i s
clear from this that a trial � provides an upper bound to the real ground
state energy� and that the value E0
is reached if and only if � � ��0
� � � C �
The fundamental minimum principle thus follows�
E��� � E0
� � � ��0
�� E��� � E0
�����
The ground state energy E implicitly de�ned by the minimization in �����
depends only on and is completely identi�ed by t h e c hoice of N and v in ������
�� ���� THE SCHR
�ODINGER EQUATION
in this sense� E is said to be a functional E�N � v � of the particle numbe r N
and of the external potentials v�
This minimum principle has a key importance in practical applications�
as an example� it is the basis of the widely used Rayleigh�Ritz variational
method� where a trial function is constructed that depends on a numbe r o f
variational parameters� that are determined by the requirement of minimiz�
ing the expectation value for the energy� O b viously� the quality of the results
depends both on the choice of the trial function and on the dimensional�
ity of the parameter space that is spanned� In addition� if a trial function
can be constructed as to be orthogonal to the n � � l o west eigenfunctions�
the Rayleigh�Ritz method provides an upper bound for the n�th eigenvalue�
The well�known Hartree ���� and Hartree�Fock ��� methods correspond to
searching the solutions for ���� respectively in the subspace of the products
of single�particle orbitals and in the subspace of antisymmetrized products
of single�particle orbitals the Slater determinants��
����� The Hellmann�Feynman theorem
There is an important consequence that can be drawn from equation �����
if the Hamiltonian H depends parametrically on some �� su ch that dH�
�d�
is well de�ned� the following theorem can be demonstrated�
Hellmann�Feynman theorem Given a Hamiltonian H�
� and a ��� sat�
isfying the Schr�odinger equation H�
� � E�
�� the following relation holds�
dE�
d � h�jH�
j�i � h�jdH�
�d�j�i
d� d�
Proof� It follows from the orthonormality property for the solutions of the
�� CHAPTER �� THEORY O F E L E C T R ONIC STRUCTURE
Schr�odinger equation� or its equivalent v ariational formulation ������ that
d
h���j���i � � � hji � � � h d�d�ji � hjd�d�i � � �
d�
Using the hermiticity o f H� it is readily shown that
d hjH�
ji � hd�d�jH�
ji � hjdH�
�d�ji � hjH�
jd�d�i �
d�
� E�hd�d�ji � hjd�d�i� � hjdH�
�d�ji � hjdH�
�d�ji
QED
The scope of the theorem is trivially extended to the expectation values of
any Hermitian operator A�
dependent on some external parameter � and for
a system prepared in a given eigenstate of A��
The original work ���� was concerned with the de�nition and the calcula�
tion of the forces acting on molecular systems in fact Hamiltonians H such
as that de�ned in ����� explicitly depend on the nuclear coordinates R��
a n d i t i s m uch more convenient to calculate directly the expectation value
of dH �dR�
than approximating this derivative with the energy di�erences4 �
Feynman�s original application of the theorem to the Hamiltonian ����� leads
immediately to the electrostatic theorem the force on a nucleus is given by a
completely classical expression� where the potential is obtained by the super�
position of the nuclear �elds and by the electrostatical potential generated
R
by the electronic charge density n�r� � N j�r1
� � � rN
�j2 dr2
� � � d rN
Z X
F�
� �r�Wn�n
� j�r1
� � � rN
�j2 r�v�ri� dr1
� � � d rN
�
i Z
� �r�Wn�n
� n�r� r�v�r� dr� �����
4 A similar problem for a time�dependent S c hr�odinger equation had already been ad�
dressed by Ehrenfest ���� in ����� namely he demonstrated a quantum mechanical equiv�
alent md2 hjrji�dt2 hj � dV �drji for Newton�s second law of dynamics�
���� DENSITY FUNCTIONAL THEORY ��
��� Density F unctional Theory
The solution to the electronic structure problem has been presented� ac�
cording to the prescriptions of the Schr�odinger equation� as the problem of
obtaining the proper ground�state many�body wavefunction �v�r1
� � � � � rN
�
for a given external potential v� A s s u c h the ground�state solution has an
intrinsically high complexity while the problem is completely de�ned by t h e
total numbers of particles N and by the external potential v�r� its solution�
uniquely determined by N and v�r� and thus a functional � � ��N � v �r� �
depends on �N coordinates� This makes the direct search for either exact or
approximate solutions to the many�body problem a task of rapidly increasing
complexity�
Soon after the introduction of the Schr�odinger equation there have b e e n
attempts to refocus the search on the ground�state charge density n�r� which
is the most natural and physical low�complexity q u a n tity in the problem� The
�rst outcome of this e�ort was the Thomas�Fermi model ��� ��� a functional
of the charge density alone was constructed from the basic assumptions of
treating the electrons as independent particles reducing the electron�electron
interaction to the Coulomb electrostatic energy and using a local density
approximation for the kinetic energy term
Z
T �n � t�n�r� dr�
where t�n�r� is the kinetic energy density for a system of non�interacting
electrons with density n� The functional that is obtained from this hypothesis
�� CHAPTER �� THEORY O F E L E C T R ONIC STRUCTURE
is5
Z Z Z Z
�
2�
5
3
2
3 n �r�dr � v�r�n�r� �
� n�r1
�n�r2
�
� jr1
� r2
j
dr1
dr2�n�r�� � ���ET
�F
and the ThomasFermi solution is de�ned as the charge density n�r� th at
minimizes this functional ET
� with the constraints of being greater than
zero and normalized to the total number of electrons N This approach has
been shown to fail in providing even qualitatively correct descriptions of
systems more complex than isolated atoms ���� its main limitation coming
from the approximate form for the kinetic energy contribution �it should be
noted that more sophisticated and accurate choices for the kinetic energy
term as a function of the electron density can be made ����� On a more
fundamental basis� the model was introduced when there was no theoretical
ground to support the choice of the charge density� rather than the many
bod y w avefunction� for the search� via the variational principle �� ��� of the
electronic ground state
F
����� The Hohenberg�Kohn theorems
The essential role that is played by the charge density in the search for the
electronic ground state was pointed out for the �rst time by H o h e n berg and
Kohn ��� in their exact reformulation of the problem that now goes under
the name of Density Functional Theory
Let us consider a �nite� isolated system of N interacting electrons in an
external potential v�r�� as described in Sec � �� with a Hamiltonian �� ���
the only additional assumption is that the groundstate � be unique� non
degenerate This latter assumption will be relaxed in � � � Again� � and
5 Neglecting for the rest of the chapter contributions from the term Wn�n�
���� DENSITY FUNCTIONAL THEORY ��
n�r� � h�jnj�i are perfectly speci�ed �albeit not explicitly� once N and�
v�r� are formulated� and thus are functionals of N � v �r�� Let now VN
�R
3�
be the class of all densities that are positive de�nite� normalized to an integer
number N � and such that there exists an external potential v�r� for which
there is a non�degenerate ground state corresponding to that density this
is the class of v�representable densities� and the following discussion6 will
be restricted to densities that belong to this class� The �rst theorem of
Hohenberg and Kohn � establishes the legitimacy of the charge density a s
the fundamental variable in the electronic problem�
�st Theorem� the Density as the Basic Variable � n�r� � V N
�R
3� the
external potential v�r� is a functional v�r� � v�r�n�r� of the charge density
n�r�� within an additive constant�
Proof� Let us assume� ad absurdum� that there exists a di�erent potential v
0
with a ground state �0 that gives rise to the same charge density n�r�� Let E
and E 0 be the respective ground�state energies taking �0 as a trial solution
�for the Hamiltonian H � w e h a ve the strict inequality
Z
� � � �E � h�0 jH j�0 i � h�0 jH
0 j�0 i � h�0 jH � H
0 j�0 i � E0 � n�r�v�r� � v
0 �r�dr
��if h�0 jH j�0 i � E then �0 would be the ground state for H � in virtue of
� ������� but the two distinct di�erential equations for H and H
0 cannot have
�the same ground state�� Similarly� taking � as a trial solution for H
0 � we
have
Z
� � � �E0 � h�jH
0 j�i � h�jH j�i � h�jH
0 � H j�i � E � n�r�v�r� � v
0 �r�dr
6See ����� for the problem of v� and N �representability o f c harge densities and density
matrices�
�� CHAPTER �� THEORY O F E L E C T R ONIC STRUCTURE
Adding together these two inequalities the absurdum is obtained�
E � E
0
� E � E
0
�
QED
It is thus demonstrated that the ground�state charge density determines� in
principle� the external potential of the Schr�odinger equation of which i t i s
solution7 and thus� implicitly� all the other electronic ground�state properties
of the system� It is then possible to introduce� in the class of v�representable
densities VN
�R
3� the functional
F �n�r� h�jTe
� Ue�e
j�i� �����
this is unique and well de�ned� and� since it does not explicitly depend on
the external potential for the electrons� it is a universal functional of the
charge density� It should be pointed out that no explicit expressions for this
functional are known to date� obviously as a by�product of the complexity o f
the many�body problem that lies at the core of the de�nition of F �n�r� �����
It is now possible to de�ne� for an arbitrary external potential v�r �as�
sumed here to be local� unrelated to the external potential implicitly de�ned
by a density i n VN
�R
3� the Hohenberg�Kohn Density Functional for the en�
ergy of the ground state�
Z
0Ev
�n �r� F �n
0�r� � v�rn
0�rdr� �����
The second theorem of Hohenberg and Kohn ��� establishes the existence of
a v ariational principle for this functional of the charge density� t h us rational�
izing the original intuition of Thomas and Fermi�
7 Precisely� it is an expectation value of the solution� since n�r� determines v�r�� which
in turn determines the ground state �� and the relation n�r� � h�j �nj�i holds�
���� DENSITY FUNCTIONAL THEORY ��
�nd Theorem� the Energy Variational Principle � n
0 �r� � V N
�R
3��
R
Ev
�n
0 �r�� � F �n
0�r�� � v�r�n
0 �r�dr � E0� w here E0
is the ground�state
energy for N electrons in the external potential v�r��
0Proof� A given n �r� determines uniquely via the st theorem its own exter�
nal potential and ground�state wavefunction �0 If this �0 is used as a trial
wavefunction for the expectation value of the Hamiltonian with the exter�
� � �j�0 i �nal potential v�r� the relation h�0 jH j�0 i � h �0 jT�
e
� Ue�e
j�0i � h�0jvR
F �n
0�r�� � v�r�n
0�r�dr � Ev
�n
0�r�� holds Now because of the minimum
0 �principle � Ev
�n �r�� � h�0 jH j�0i � E0
Since the ground state is non�
degenerate the equality holds only if �0 is the ground state for the potential
v
QED
These two theorems show that the problem of solving the Schr�odinger equa�
tion for the ground state can be exactly recast into the variational problem of
minimizing the Hohenberg�Kohn functional � � with respect to the charge
density� the complexity of the problem is reduced in principle from having
to deal with a function of �N variables to one that depends only on the �
spatial coordinates
����� The Levy approach
A simpler and more direct proof for the Hohenberg�Kohn theorems has been
introduced by Levy ����� this has the advantage of removing the require�
ment of non�degeneracy for the ground state of the electronic problem while
focusing directly on the minimization procedure in the de�nition of the func�
�� CHAPTER �� THEORY O F E L E C T R ONIC STRUCTURE
tional itself� In addition� the ill�de�ned problem of the representability o f
the charge density is greatly simpli�ed �see ������
Levy ������ introduced a new de�nition for functionals of the density�
based on the concept of constrained s e arch� Given an operator A and a
density n�r� the functional An�r� is de�ned as the minimum expectation
value of A for a constrained search along all the many �bod y a n tisymmetric
wavefunctions for which h�jnj�i � n�r�
An�r� � minh�jAj�i ������!n
This de�nition is meaningful �the set is non�empty for those densities that
do admit a representation derived from a N �bod y w avefunction� it is the class
of N �representable densities� much simpler to characterize than the class of
v�representable densities �see ������ The set is also known to be bounded
from below ���� The Levy proof develops as follows8 �
Theorem� the Levy proof The ground�state electron density n0�r min�R
imizes the functional Ev
n�r� � F n�r� � v�rn�rdr� w here F n�r� is a
universal functional of n�r� the minimum value is the ground�state electronic
energy E0�
Proof� Let the operator F be Te
� Ue�e
� F n�r� is de�ned uniquely �and
it is a universal functional of the charge density alone via the constrained
search m in �!n
h�jF j�i� For an arbitrary n�r� let �n
be the N �electron
wavefunction that yields the charge density n�r and minimizes h�jF j�i�
R
Ev
n�r� is F n�r� � v�rn�rdr� and so it follows from the de�nition of �n
that Ev
n�r� � h�n
jF � v j�n
i� but this is the Hamiltonian in the presence
8 As a matter of clarity� a subscript is introduced to denote all the quantities that refer
to the ground state ��0
� n 0� E 0��
�� ���� DENSITY FUNCTIONAL THEORY
of the external potential v�r�� and so we h a ve� from the minimum principle
������ that for every N �representable charge density
Ev
n�r� � E0
�
Now it remains to be demonstrated what is the value of Ev
evaluated on
the ground state charge density n0
� Since the ground�state wavefunction �0
integrates to the ground�state charge density n0
� it certainly takes part in
�the constrained search that de nes F n0�r�� Thus� F n0�r� � h �0
jF j�0
i�
and so� adding the external potential term on both sides� we h a ve
Ev
n0�r� � E0
�
that� together with the previous relation� gives us Ev
n0
�r� � E0�
QED
It should be mentioned that� although very powerful� this approach misses
one of the elegant results of Hohenberg and Kohn� namely that there are no
two di�erent external potentials �a part from some additive c o n s t a n t� that
give rise to the same ground�state electronic charge density�
����� The representability problem
The original Hohenberg�Kohn proof takes place in the space of charge densi�
ties that are v�representable� again� this means that they are positive de nite�
normalized to an integer number� and such that there exists an external po�
tential v�r� for which there is a non�degenerate ground state corresponding
to that density� In this framework� the Hohenberg�Kohn �st theorem estab�
lishes a one�to�one mapping between a v�representable charge density a n d
�� CHAPTER �� THEORY O F E L E C T R ONIC STRUCTURE
the external potential for which this density is the the expectation value of
the ground state� and in this space VN
�R
3� the universal functional F and the
variational principle of the �nd theorem are formulated� The characterization
of the class VN
�R
3� is still lacking� v�representability has been demonstrated�
in a lattice version of the Schr�odinger equation� for all the particle�conserving
charge densities that are in the neighbour of a v�representable density ���
This� together with the more serious necessity to adopt approximations for
the universal functional F � has lead to a generalized application of the varia�
tional principle for all of its practical implementations� It should be pointed
out that simple counter�examples of densities that do not belong to VN
�R
3�
have been presented ���� �� e�g� if a Hamiltonian has a ground state with
degeneracy greater than �� the density obtained from a linear average of each
ground state is in general not v�representable ����
The Levy proof overcomes this problem� by requiring the density j u s t t o
be obtainable from some antisymmetric wavefunction �in addition to being
positive de�nite and normalized to an integer number�� in order to allow for a
meaningful constrained search� This is the condition of N �representability� a
much w eaker requirement that is satis�ed by a n y w ell�behaved charge density�
In fact� it has been demonstrated by Gilbert ��� that the only condition
required is proper di�erentiability�
Z
1
2jrn �r�j2dr � R�
to ensure that the kinetic energy of the auxiliary orbitals used in the con�
struction of an antisymmetric wavefunction from a given n�r� remains �nite�
���� DENSITY FUNCTIONAL THEORY ��
����� Density operators
Up to now� a quantum state has been described in terms of its N�body
wavefunction ��r1
� � � � � rN
�� with the expectation value for a given observable
� � �A �Hermitian operator� given by hAi � h�jAj�i A more general description
can be obtained with the introduction of the density operator �N
� j�ih�j��
whose representation in the coordinate space is the density matrix
0 0���r � � � � � rN
���r1
� � � � � rN
��1
R
For a normalized �� tr ���N
� � ���r1
� � � � � rN
���r1� � � � � rN
� dr1
� � � d rN
� �
�N
is clearly idempotent� and thus it is a projection operator The expecta��
tion values can now be expressed as a trace on this operator� in the coordinate
representation and for a normalized � we h a ve
� � � �� �hAi � tr � �N
A� � tr � A�N
� � h�jAj�i� ���
The description of a system at a nite temperature requires the introduction
of an ensemble density operator� since then the system is part of a much larger
closed system �a heat bath with quasi�in nite thermal capacity� for which a
complete Hamiltonian description is unattainable In this case� the system is
said to be in a mixed s ta te � where it cannot be characterized by a w avefunc�
tion �as opposed to the pure states of all the preceding discussion�� but only
�as a mixture over all pure states The ensemble density operator �N
is thus
�
P
de ned as �N
�
i
wij�iih�i
j� where the sum is taken over all the available
pure states The wi
are often described as the probabilities of nding the
system in the relative pure state� although this is conceptually misleading
����� and the wi
are just a statistical distribution outcome of both the prob�
abilistic interpretation of the wavefunction and of the statistical description
�� CHAPTER �� THEORY O F E L E C T R ONIC STRUCTURE
coming for our imperfect knowledge of the full system� This becomes appar�
ent in the statistical generalization of ������ for the expectation value of an
observable for a mixed state� which i s g i v en by
X
� � � �� �hAi tr � N
A� tr � AN
� wih�ijAj�ii� ������
i
where the cross terms that would arise if the mixed state was actually a
superposition of pure states are not included� In contrast to the density
operator� the ensemble density operator is no longer idempotent they are
both Hermitian and uniquely de�ned �as opposed to the wavefunction� that
has an arbitrary phase attached�� An important relation follows from the
� �de�nitions and from the time�dependent S c hr�odinger equation i~j�i H j�i� � � � � � �� � � H H
��N
j�i h�j � j�i h�j j�ih�j � j �ih�j �
�t �t �t i~ i~
from which t h e Liouville equations for the density operator and� by the linear�P�ity of N
wij�iih�i
j� for the ensemble density operator� are obtained�
i
� �
� � � ��i~ �N
� H � �� N
�� i~ N
� H� N
�� �������t �t
The Hamiltonian and the ensemble density operator commute in the case of
a stationary state�
��H
� � N
� � � ������
and a complete system of eigenvectors for both of them can be found� This
is a fundamental relation that will be fully exploited in the development o f
ca sea practical algorithm �albeit for the slightly di�erent o f n o n �i n teracting
electrons in a self�consistent potential��
It is possible and useful to de�ne a reduced density matrix of pth order�
which in the coordinate representation has the following form� Z Z
0 0
N �
0 0�p
�r1���r p
r1���rp� ��� ���r1
���rN
���r1
���rN
� drp+1���drN
�
p� � N � p��
���� DENSITY FUNCTIONAL THEORY ��
�Since the Hamiltonian H in ����� is a function of one�particle and two�particle
terms only� it can be demonstrated that its expectation value can be written
as an explicit functional of the density matrices of order � and � �� �Sec� �����
or� since �1
descends from �2
� just of the density matrix of order ��
�E tr � H��N
� E�1
� � 2
� E�2� ������ Z Z Z
�
2 �� r � v�r�� �1
�r� r� dr �
�
��
2
�r1� r2
� r1� r2
� dr1
dr2
�
jr1
� r2
j
It is possible to construct a density�matrix functional theory and demon�
strate a variational principle for the functional in ������ ���� with the eigen�
vectors of �1
�the natural orbitals� and �2
�the natural geminals� as the
search v ariables� and reducing again the complexity of the problem from
a �N�dimensional search to a ��dimensional search� The great obstacle is
that �1
and �2
must be N �representable� thus descending from an appropri�
ate antisymmetric �� and this characterization �especially for �2
� is still
9
unsolved ����� It is nevertheless interesting to contrast this problem of min�
imizing an explicit functional in a class of N �representable density matrices
that escapes characterization� with the density�functional problem of mini�
mizing a functional that is de�ned only implicitly in the well�de�ned simple
class of N �representable densities�
9Note that �2
depends on � independent v ariables� but� because of density�functional
theory� is completely characterized by n�r�� This means that the set of N�representable
density matrices of order � has zero Lebesgue measure�
�� CHAPTER �� THEORY O F E L E C T R ONIC STRUCTURE
����� Canonical�ensemble formulation
From the fundamentals of classical statistical mechanics we h a ve that the
entropy i s
S�E � V � � kB
ln ��E�� ������
where ��E� is the volume in the phase space occupied by the microcanonical
ensemble at energy E� The same de�nition holds for quantum mechanics
with ��E� computed properly via a sum over all the states of the system
X
��E� � W �fwig � ������
fwi
g
w
where W �fwig is the number of states available for a given set of occupations
i
of the accessible quantum states and the sum is done on all the possible
sets that are compatible with the thermodynamical ensemble� In the statis�P�tical description via the ensemble density operator �N
�
i
wij�iih�i
j� the
distribution is over a set of pure states j�ii with the wi
following the rules of
P
a distribution �wi
� � and wi
� ��� The entropy can thus be equivalently
i
written as
X
� �S�E � V � � �kB
wi
ln wi
� �kB
tr ��N
ln �N
�� ������
i
where the second equality exploits the independence of the value of the trace
of an operator from the representation in which it is expressed�
The choice of the ensemble �canonical in this case� determines the equi�
librium ensemble density operator by requiring that it maximizes the entropy
P
while keeping the constraints on the statistical distribution
i
wi
� � and
�on the expectation value for the energy tr��N
H� � E� These conditions lead
���� DENSITY FUNCTIONAL THEORY ��
�0to the equilibrium canonical ensemble density operator �N
���� �Sec� ���
^��H
� e
�0 �
� � �N
^tr e
��H
Following � ��� the Helmholtz free energy A � E � TS for a given ensemble
density operator can be introduced � �
� � �A��N
� � tr �N
ln �N
� H � � �� �
Now a v ariational principle can be demonstrated� that is the exact equiv�
alent of the Rayleigh�Ritz variational principle � �� for the case of �nite
�temperature� namely that for every positive de�nite �N
with unit trace
A��N
� � A��0 �� � ���N
0where � N
is the ground�state ensemble density operator� The proof relies P
on Gibbs� inequality � wi�ln wi
� ln w
0 � � and on Jensen�s inequality i i
� ��hexp�Ai � exp�hAi and can be found in Ref� ���� �Sec� �� �
Once the variational principle is formulated� the demonstration of the
existence of a �nite�temperature� canonical density functional theory follows
the steps of Sec� ��� either along the original Hohenberg�Kohn proof ���� or
in a constrained search f o r m ulation� In brief� following the Levy approach� a
�canonical ensemble universal functional of the charge density n�r can be
uniquely de�ned� via the constrained search on all the density operators that
integrate to n�r � � ��
� � � � n rF�
� � � � min tr �N
Te
� Ue�e
� ln �N
� ����̂N
!n
�
With this universal functional at hand� the canonical Mermin density f u n c �
tional is written as Z
Av
�n�r� � F��n�r� � v�rn�r dr�
�� CHAPTER �� THEORY O F E L E C T R ONIC STRUCTURE
Now� for a given n�r�� a �N
that minimizes ������ is dened� and a variational
principle for the Mermin functional holds �from Eq� �����
Av
�n�r�� A��N
� � A��0
N
�� ������
The Mermin functional evaluated on the ground�state charge density n0
�r�
involves a search on all the ensemble density operators that integrate to n0�r��
and thus includes the ground state �0 � it follows that Av
�n0�r�� � A��0
N N
��
that� combined with the variational principle for the Mermin functional� gives
Av
�n0�r�� A��0
N
��
��� The Kohn�Sham mapping
Density F unctional Theory provides the theoretical ground for reformulating
the ground�state many�body problem as a variational problem on the charge
density� the constrained minimization on the Hohenberg�Kohn density func�
tional for N electrons can be rewritten� introducing the indeterminate mul�
tiplier �� as the variational problem
� � �� Z Z
� F �n�r�� � v�r�n�r� dr � � n�r� dr � N � � ������
Formally ������ leads to the Euler�Lagrange equation for the charge density10
�F �n�r��
� v�r� �� �������n �r�
10 The derivatives with respect to the charge density are always de�ned a part an additive
R
constant� since K �n �r� dr � �� this can be absorbed into the chemical potential ��
Otherwise� see Sec� ����� and Refs� �� and � for the characterization of the domain
for the variations of the charge density�
���� THE KOHN�SHAM MAPPING ��
The two approaches of minimizing directly the Hohenberg�Kohn density f u n c �
tional ������ or solving for the associated di�erential equation ����� are
equivalent11 although they might di�er in the computational e�ort associ�
ated with their implementation�
With these results in mind the Thomas�Fermi method can be reinter�
preted as a tentative approximation for the unknown universal functional
F �n�r�� taken to be the sum of the classical part of the electron�electron
interaction plus the kinetic energy of the non�interacting electron gas in the
local density approximation� This approach and its subsequent re nements
although very economical have s h o wn only moderate quantitative agreement
with experimental data �see Ref� ���� for a review and Ref� ���� for recent
more accurate developments��
A m uch i m p r o ved strategy has been developed by Kohn and Sham ���
that successfully readdressed the problem of nding a better approximation
to the kinetic energy albeit at the expense of having to deal with an addi�
tional set of auxiliary orthonormal orbitals� The kinetic energy is a one�body
operator and as such the expectation value �see Eq� ����� can be written in
terms of the reduced density matrix of order ��
Z 1 X � �
T � �
�
r
2�1
�r0� r��
r0 �r
dr � � wi
h�i
jr2j�i
i�
� �
i�1
where the last equality comes from expressing the density matrix in the
representation of its natural orbitals �i
� In general the natural orbitals are
in nite in numbe r � H o wever if N non�interacting electrons are considered
their ground state antisymmetric wavefunction can be expressed as the Slater
11 Note that ������ identi�es the minimum only if there are no other extremal points� see
Ref� �� for a discussion and ��� for a proof that the exact F is indeed a convex functional�
�� CHAPTER �� THEORY O F E L E C T R ONIC STRUCTURE
determinant of only N orbitals� which are the lowest eigenstates of the non�
interacting Hamiltonian�
The fundamental assumption of Kohn and Sham is to introduce a refer�
ence system of non�interacting electrons in an external potential v KS
�r� such
that the ground state charge density for this problem is the n�r� that enters
the Hohenberg�Kohn functional� This de�nition is meaningful �at least in its
original formulation� if the charge density is non�interacting v�representable�
so that there can be a unique de�nition for the total energy Kohn�Sham
R
functional Ts�n�r� v KS
�r�n�r� dr� where Ts
is now the kinetic energy
associated with the new reference system� This technical problem can be
overcome� again� by de�ning the kinetic energy part of the Kohn�Sham func�
tional via a constrained search o ver all the Slater determinants of order N
that integrate to a given charge density n�r��
� � ZN
��Ts�n�r� min h�S
jTej�S
i min
X
�
�
�i
�r�r
2�i�r� dr �
�S
!n �S
!n �
i�1
The Ts�n�r� can then be generalized by extending the search to all the an�
tisymmetric wavefunctions that yield a charge density n�r� �whether it is
non�interacting v�representable or not� and it can be demonstrated ��� that
this generalized Ts
coincides with the original de�nition for all the densi�
ties that are non�interacting v�representable� With this de�nition at hand�
a further decomposition of the universal functional F can take place via the
introduction of the classical electrostatic energy term EH
�n�r��
F �n�r� Ts�n�r� EH
�n�r� Exc
�n�r��
Z Z
�
EH
�n�r�
�
n�r1�n�r2
�
dr1
dr2
�
jr1
� r2j
�� ���� THE KOHN�SHAM MAPPING
This decomposition� where F � Ts
and the electrostatic energy are well�de�ned
quantities� acts as an implicit de�nition for the exchange�correlation energy
Exc� that collects the contributions from the non�classic electrostatic inter�
action and from the di�erence between the true kinetic energy T and the
non�interacting one Ts� The real success of the Kohn�Sham reformulation
lies ultimately in the fact that Exc� which is the term where all the in�
escapable complexity o f t h e m a n y�body problem has been �nally pushed� is
a small fraction of the total energy and� more importantly� that it can be
approximated surprisingly well� These approximations� discussed in Chap� ��
are at present the strength of Density F unctional Theory�in itself a very
ecient reformulation for the quantum�mechanical problem�and its limit�
in the very good but not exact approximation they provide�
The Euler�Lagrange equation ���� is rewritten with the new terminology
as
�T s�nr �
� v KS
r � �� ���� �n r
where the e�ective p o t e n tial is
Z
nr0 �E xc�nr �
v KS
r � vr � dr
0 � vxcr � vxcr �
r � r0j j �n r
and� again� it is such that the set of non�interacting electrons with kinetic
energy Ts
in its �eld have the same ground�state charge density a s t h e i n ter�
acting electrons in the external potential v� The e�ective potential is now a
function of the charge density itself� the problem has become self�consistent�
and the solution for the reference system of non�interacting N�electrons
�
�S
� pN �
det��1� � � � � � N
� ����
�� CHAPTER �� THEORY O F E L E C T R ONIC STRUCTURE
is now bound to produce a charge density that consistently outputs the orig�
inal input e�ective p o t e n tial�
Ts�n�r�� as such is still an unknown functional of the density n�r� but now
it can be easily �but more expensively� written in terms of the N orbitals �i
for the non�interacting electrons that act as auxiliary variables and properly
span the domain of all the N �representable densities �provided that they are
continuous and square integrable�� The exact kinetic energy for the Kohn�
Sham reference system is straightforwardly
N X�
Ts�n�r�� � � h�i
jr2j�ii� ���� ��
i�1
It should be noted that the �i
have to be orthonormal �solutions for the
non�interacting fermion problem� in order for the kinetic�energy expression
���� � to be valid�
The variational problem on n�r� i s t h us �nally reformulated in terms of
a constrained search on the N � i where N
2 Lagrange multipliers �ij
are
introduced to provide for the orthonormality of the orbitals �and thus for the
charge conservation�
Z
�
N
� �� X
� Ts�n�r�� � vK S
�r�n�r� dr � �ij
h�j
j�ii � I � � � ������
i�j�1
From this a single�particle Schr�odinger�like equation is derived
N X�
2�� r � vKS
� �j
� �ij
�j
� �������
i�1
�ij
is a Hermitian matrix which can be diagonalized via a unitary transforma�
tion of the orbitals that leaves the non�interacting N�electron wavefunction
������ invariant and consequently the density and the e�ective Hamiltonian
���� THE KOHN�SHAM MAPPING ��
2
2
S
�
1 r � vK
� F rom this� the �canonical� Kohn�Sham equations follow12 � � �
�
�
r
2 � vH
r� � vxcr� � vextr� �ir� � HKS
�ir� � �i
�ir� ���� Z
nr0 � �E xc
vH
r� � dr0 � vxcr� �
jr � r0 j �n r�
N X
nr� � j�ir�j2�
i�1
These equations should be iterated to self�consistency� u n til the minimum
for the Kohn�Sham functional is reached� Kohn and Sham proposed taking
the lowest N �orbitals to form the determinantal many�body wavefunction�
and this choice works consistently for all the charge densities that are non�
interacting v�representable that is� for which Ts
can be written in terms of
single�particle orbitals��
The Kohn�Sham equations are the Euler�Lagrange equations for the con�
strained minimization of the Kohn�Sham functional� It should be noted that
the search for the ground state can also proceed via the direct minimization
of the full functional ZN
�E�f�ig� �
X
�
�
�i
r�r
2�ir� dr � EH
�nr�� �
i�1 Z
�Exc�nr�� � vextr�nr� dr �����
with respect to the N auxiliary orbitals �i� with the proper constraints of
orthonormality h�ij�j
i � �ij
and charge density conservation� An alternative
expression for the total energy can be derived by extracting the band�energy � � P
term
i
�i
� tr T
s
� vKS
�
ZN X
E�f�ig� � �i
� EH
�nr�� � Exc�nr�� � vxcr�nr� dr� �����
i�1
12A subscript is added for clarity to the external potential v � vext�
�� CHAPTER �� THEORY O F E L E C T R ONIC STRUCTURE
����� Fractional occupancies� Janak�s theorem
An extension of the previous formalism that includes the case of fractional
occupancies has been proposed by Janak ����� Janak de�nes a generalized
functional of M �� N orbitals f�i
g and of their occupation numbe r s ffi
g
in the language of the constrained search formulation the functional is � �
M
Z
� � X
TJ
�n�r� � P
min fi
�i��r �
�
r
2 �i�r dr � ����
fi
j�i
j2 �n �
i�1
where the search i s p e r f o r m e d o ver all orthonormal orbitals and occupa�P
tion numbers that yield the density n�r �h en ce fi
� N � The condi�i
tion � � fi
� � �i assures that the density matrix of order � is ensem�
ble N �representable ����� and so it derives from an appropriate �N
conse�
quently Eq� ����� provides the variational principle to develop a density�
functional implementation in strict analogy with the Kohn�Sham discussion�
The Janak�s functional is formulated�
M
Z
� � X
E�ffig� f�ig� � fi
�i��r �
�
r
2 �i�r dr � EH
�n�r� �
�
i�1 Z
�EJ�xc
�n�r� � vext�rn�r dr� �����
where EJ�xc
is rede�ned to take i n to account the new de�nition for the non�
interacting kinetic energy� and where now the minimization is performed with
respect to the f�ig and the ffig� instead of the determinantal wave function
����� of the Kohn�Sham case� For a �xed set of ffig the orbitals f�ig must
satisfy the corresponding Euler�Lagrange equation� �
M X ��
E�ffig� f�ig� � �i
� �h�ij�ii � � � � � i�e�
�� i�
i�1 � �
��
�
fir
2 � fivJ�KS
�i
� ��
i�i� ������
�� ���� THE KOHN�SHAM MAPPING
where now the Lagrange multipliers are diagonal� and the orthogonality
comes from the previous di�erential equation ������ �this is due to the choice
of the TJ
functional as a diagonal sum�� These are again KohnShamlike
equations� and obviously for fi
� � �i
�0 �fi�i
The dependence of the functional E on the occupation numbers can be
seen by calculating the unconstrained variation of the energy with respect to
one fi� �E ��f i� while allowing the orbitals to relax�
Z Z
�E �
�i
�r�r
2�i�r� dr
��vH
vxc
vext� �n �r� �
� dr
�f i
� �n �r� �f i
Z Z
� � � �i
�r�r
2�i�r� dr vJ�KS
�i��r��i�r� dr �i
�������
The expression above is referred to as Janak�s theorem� The constrained
variation of the energy functional with respect to fi� that is the unconstrained
minimization of E � �N� gives �����
X
��E � �N � ��i
� ���f i� ������
i
which p r o vides a rationale to the choice of fi
� or � for �i
greater or smaller
than � �so that ��E � �N � � ��� It is apparent from Eq� ������ that Janak�s
functional is not variational with respect to the occupation numbers �to a
variation in the occupation numbers� precisely�� as noted in ���������
Finally� it should be mentioned that there are some conceptual problems
in the foundations of Eq� ����� as noted in Ref� ����� Sec� �� and more re
cently in Ref� ����� The objections arise in considering occupation numbers
di�erent from either � or � in zerotemperature densityfunctional theory�
and in fact there are indications �������� that the domain of de�nition for the
functional derivatives with respect to the charge density might not include
�� CHAPTER �� THEORY O F E L E C T R ONIC STRUCTURE
charge densities written in terms of fractional occupation numbers� Either
on these methodological grounds� or following Janak�s analysis� the conclu�
sion is that occupation numbers �with an exception made for orbitals which
will eventually be degenerate at the Fermi level� are bound to assume at
self�consistency a v alue that is either � or � Fractional occupation numbers
should thus be properly considered only in a statistical sense� as the output
of a nite�temperature formulation�
�� ���� THE KOHN�SHAM MAPPING
����� Finite�temperature extension
The extension of density functional theory to the canonical ensemble has
been discussed in Sec� ������ where the variational principle for the canonical
Mermin functional Av
�n�r has been demonstrated� following from Eq� �����
A Kohn�Sham mapping onto single�particle orbitals in a self�consistent p o �
tential will be discussed here� �rst for the case of non�interacting electrons
and then for interacting ones�
If the electrons are non�interacting� the allowed states are the solutions
of the single�particle equations
� �
�
2� r v� ext
�i
� �i�i� ������
The occupation of each state is determined by the statistics �that is� ulti�
mately� a prescription for counting the states and by the temperature� the P
requirement that the entropy kB
ln ��E � ffi
g
W �ffig is maximized� to�
gether with the Fermi�Dirac statistics� brings in the familiar Fermi�Dirac
distribution ����
� �
�i
� � �
fi
� f � � �����T � exp�
�i
�� T
for this distribution the entropy S � kB
ln ��E is
X
S � �kB
�fi
ln fi
�� � fi ln �� � fi �����
i
and the expectation value of the kinetic energy operator is�
� � Z Z X
Ts
�
X
fi
�i��r �
�
r
2 �i�r dr � fi�i
� vext�rn�r dr� ������
i i
These expressions can be derived more formally with the help of second�
quantization operators �see Ref� ���� Chap� ��
�� CHAPTER �� THEORY O F E L E C T R ONIC STRUCTURE
The Kohn�Sham formulation for a system of interacting electrons �see
Refs� ������������� follows the lines of the zero�temperature case a refer�
ence system of non�interacting electrons is introduced for which the ground
state has the charge density n�r� that enters in the canonical Mermin func�R
tional Av
�n�r�� � F�
�n�r�� � vext�r�n�r� dr� An exchange�correlation po�
tential E� �xc
�n�r�� is then implicitly de ned by isolating in F��n�r�� the non�
interacting kinetic energy T and entropy S of the non�interacting system
de ned in ������ and ������� The minimization of the Mermin functional
with the constraint o f c harge normalization leads to the canonical Mermin�
Kohn�Sham equations � �
�
2� r � vH
�r� � v� �x c
�r� � vext�r� �i�r� � �i
�i�r� ������� Z
vH
�r� �
n�r0 �
dr
0 � v��xc�r� �
�E ��xc
jr � r0 j �n �r� � � X �i
� �
X
fij�i�r�j2�f � N� n�r� �
T
i i
These equations are formally similar to the zero�temperature case although
now there is a Lagrange multiplier � to constrain the sum of the occupa�
tions to the total number of electrons N since the number of states that
the nite�temperature brings in jumps from N to � �although for all pur�
poses one can neglect all the states higher than a threshold for which the
occupancies are practically zero�� Alternatively one can minimize directly
the canonical Mermin functional as a function of the ffig and f�ig w ith the
explicit constraints of charge normalization and orthonormality � � Z X
A � T � f�ig� ffig� � fi
�i��r� �
�
r
2 �i�r� dr�EH
�n�r���E��xc�n�r����
i � � Z X
� vext�r�n�r� dr � T kB
�fi
ln fi
� �� � fi� ln �� � fi�� � ������
i
�� ���� THE KOHN�SHAM MAPPING
An alternative expression can be obtained introducing the band�energy term�
X
A � T � f�i
g� ffi
g� � fi
�i
� EH
�n�r� E� �xc
�n�r�
i � � Z X
� v� � x c�rn�r dr T kB
�fi
ln fi
� � � fi l n � � � fi� ����
i
A di�erent and very interesting approach has been proposed ���� �see also
�� �� in which the kinetic�energy is not determined via a Kohn�Sham based
orbital representation� but it is propagated a�la Feynman from high temper�
atures� Starting from the grand�canonical potential for the non�interacting h i
� �electrons � � �
1 ln det I exp����H � � � where H is the single�particle
�
� �Te
vext� a Kohn�Sham decomposition is performed� The minimization of
h i� �A�n�r� T � � � ln det I exp����H � �
� Z
�EH
�n�r� E��xc�n�r� � v� �xc
�rn�r dr �����
is then reached by iterating to self�consistency with the constraint o f c harge
density normalization� the density matrix is evaluated as a path integral
�
��H
^
�
��H
^ �P
�P
e e �
employing the Trotter decomposition� and with the choice of a real�space
propagator for both the external potential and the kinetic energy operator�
With this real�space formulation� this approach does not su�er from errors
in the Brillouin Zone sampling of the kinetic�energy or of the charge density�
although it does require a �ner representation in real�space� In the present
case� the minimization is reached iteratively� and not in a strictly variational
fashion� In order to gauge the relative a d v antages and drawbacks a set of
applications to a wider class of systems is needed� the relevant issues are
�� CHAPTER �� THEORY O F E L E C T R ONIC STRUCTURE
the relative costs of an orbital�based formulation in comparison to a charge�
density one� the cuto� requirements for the chosen representation �e�g� plane
waves vs� a real space grid�� and the added computational stability that a
strictly variational formulation does provide�